Mixture composite regression models with multi-type feature selection

The aim of this paper is to present a mixture composite regression model for claim severity modelling. Claim severity modelling poses several challenges such as multimodality, heavy-tailedness and systematic effects in data. We tackle this modelling problem by studying a mixture composite regression model for simultaneous modeling of attritional and large claims, and for considering systematic effects in both the mixture components as well as the mixing probabilities. For model fitting, we present a group-fused regularization approach that allows us for selecting the explanatory variables which significantly impact the mixing probabilities and the different mixture components, respectively. We develop an asymptotic theory for this regularized estimation approach, and fitting is performed using a novel Generalized Expectation-Maximization algorithm. We exemplify our approach on real motor insurance data set

Hypothesis Testing in Finite Mixture Models

Mixture models provide a natural framework for unobserved heterogeneity in a population. They are widely applied in astronomy, biology, engineering, finance, genetics, medicine, social sciences, and other areas. An important first step for using mixture models is the test of homogeneity. Before one tries to fit a mixture model, it might be of value to know whether the data arise from a homogeneous or heterogeneous population. If the data are homogeneous, it is not even necessary to go into mixture modeling. The rejection of the homogeneous model may also have scientific implications. For example, in classical statistical genetics, it is often suspected that only a subgroup of patients have a disease gene which is linked to the marker. Detecting the existence of this subgroup amounts to the rejection of a homogeneous null model in favour of a two-component mixture model. This problem has attracted intensive research recently. This thesis makes substantial contributions in this area of research. Due to partial loss of identifiability, classic inference methods such as the likelihood ratio test (LRT) lose their usual elegant statistical properties. The limiting distribution of the LRT often involves complex Gaussian processes, which can be hard to implement in data analysis. The modified likelihood ratio test (MLRT) is found to be a nice alternative of the LRT. It restores the identifiability by introducing a penalty to the log-likelihood function. Under some mild conditions, the limiting distribution of the MLRT is 1/2\chi^2_0+1/2\chi^2_1, where \chi^2_{0} is a point mass at 0. This limiting distribution is convenient to use in real data analysis. The choice of the penalty functions in the MLRT is very flexible. A good choice of the penalty enhances the power of the MLRT. In this thesis, we first introduce a new class of penalty functions, with which the MLRT enjoys a significantly improved power for testing homogeneity. The main contribution of this thesis is to propose a new class of methods for testing homogeneity. Most existing methods in the literature for testing of homogeneity, explicitly or implicitly, are derived under the condition of finite Fisher information and a compactness assumption on the space of the mixing parameters. The finite Fisher information condition can prevent their usage to many important mixture models, such as the mixture of geometric distributions, the mixture of exponential distributions and more generally mixture models in scale distribution families. The compactness assumption often forces applicants to set artificial bounds for the parameters of interest and makes the resulting limiting distribution dependent on these bounds. Consequently, developing a method without such restrictions is a dream of many researchers. As it will be seen, the proposed EM-test in this thesis is free of these shortcomings. The EM-test combines the merits of the classic LRT and score test. The properties of the EM-test are particularly easy to investigate under single parameter mixture models. It has a simple limiting distribution 0.5\chi^2_0+0.5\chi^2_1, the same as the MLRT. This result is applicable to mixture models without requiring the restrictive regularity conditions described earlier. The normal mixture model is a very popular model in applications. However it does not satisfy the strong identifiability condition, which imposes substantial technical difficulties in the study of the asymptotic properties. Most existing methods do not directly apply to the normal mixture models, so the asymptotic properties have to be developed separately. We investigate the use of the EM-test to normal mixture models and its limiting distributions are derived. For the homogeneity test in the presence of the structural parameter, the limiting distribution is a simple function of the 0.5\chi^2_0+0.5\chi^2_1 and \chi^2_1 distributions. The test with this limiting distribution is still very convenient to implement. For normal mixtures in both mean and variance parameters, the limiting distribution of the EM-test is found be to \chi^2_2. Mixture models are also widely used in the analysis of the directional data. The von Mises distribution is often regarded as the circular normal model. Interestingly, it satisfies the strong identifiability condition and the parameter space of the mean direction is compact. However the theoretical results in the single parameter mixture models can not directly apply to the von Mises mixture models. Because of this, we also study the application of the EM-test to von Mises mixture models in the presence of the structural parameter. The limiting distribution of the EM-test is also found to be 0.5\chi^2_0+0.5\chi^2_1. Extensive simulation results are obtained to examine the precision of the approximation of the limiting distributions to the finite sample distributions of the EM-test. The type I errors with the critical values determined by the limiting distributions are found to be close to nominal values. In particular, we also propose several precision enhancing methods, which are found to work well. Real data examples are used to illustrate the use of the EM-test

Species richness estimation for benthic data

This thesis addresses species richness estimation for benthic data by describing the clustering of individuals within a species using a Neyman Type A distribution, and incorporating this into species richness estimates. A review of current species richness estimation methods is included. The maximum-likelihood approach to species richness estimation is extended to incorporate the Neyman Type A model, with a gamma mixing distribution on the mean abundance of individuals within a species. Species richness estimates of this model are compared to those of the simpler negative binomial and Poisson models. The use of a penalised-likelihood is applied to avoid spuriously large estimates of species richness that can be associated with the "boundary problem". The Bayesian approach to species richness is considered, using uninformative and informative priors. Informative priors are elicited using expert opinion obtained from a number of benthic ecologists at the Centre for Environment, Fisheries and Aquaculture Science. These are incorporated into species richness estimation in the form of priors, and also converted into penalties for use in the frequentist approach. Several benthic data sets are analysed throughout, along with a Lepidoptera data set, and a data set from a common bird census carried out in the USA. In addition, several simulation studies are undertaken to illustrate the performance of the estimators. The research culminates in the application of species richness estimators to estimate species mortality due to dredging carried out off the Norfolk coast. Several estimators can be considered to gain a picture of the effect of dredging, and I recommend that species richness estimators should reflect the underlying distribution of the data. I also recommend that a precautionary approach should be taken when using these estimators in practical applications